Pattern Generation

We have begun to show students how to manipulate space to construct specific figures. It is important for the students to recognize that many of the geometric figures exist in nature and in the world around them. Let us first instruct students to manipulate squares. A very simple pattern develops. (see figure 4). Three dimensional cubes may also be created when squares are joined together. (figure available in print form)

Figure 4

More interesting are the patterns and forms that are generated when equilateral triangles are arranged. Two patterns emerge, a pattern in which the triangles are adjacent to one another to form, more or less, a line, and a pattern emerges when six triangles are arranged around a center point, a hexagon appears. (figure 5)

(figure available in print form)

repeating triangles

hexagon

Figure 5

These are clearly 2-dimensional patterns that are identifiable in many tile patterns seen in kitchen floors, tile floors, stain glass windows and in ornamentation on buildings.

If we remove one triangle and arrange those 5 triangles around a central point, the only possible construct is a 3-dimensional cup shape. This in the only pattern that can be constructed. The figure will not lie in an 2-dimensional plane. (see figure 6) Seven triangles centered around the center point would result in a curved or saddled shape. (Lesson 4) No other shapes are available when the triangles are adjoined. The three shapes, the 2-dimensional linear pattern of hexagons, the cupped shape of pentagons, and the undulating saddle with 7 triangles, are the only possible constructions.

(figure available in print form)

5 triangles (pentagon) to form a cup shape

Figure 6

Students can manipulate these triangles to discover whether any other patterns can emerge. All other patterns can be reduced to these cubic, hexagonal, or pentagonal shapes. It is also interesting to note that the same geometric shapes or systems can be derived by looking at other collections of shapes. Take, for example, a set of circles placed in a repeating pattern as in figure 7. If we connect the centers, our square pattern will appear.

(figure available in print form)

square system

hexagonal system

Figure 7

Figure 8

When circles are arranged as in figure 8, the hexagonal pattern will appear. Should we stack spheres instead of circles, and connect the centers as shown in figure 10, the tetrahedron would emerge.

(figure available in print form)

tetrahedral 4 point system

(figure available in print form)

Figure 10

It is amazing to see how the same systems and patterns repeat themselves in nature. Students should play and build models to help them understand how these systems appear by themselves, not by some unnatural application of mathematics or science.

Natural Patterns

These same patterns can be found in nature. For the young scientist, it is known that all living things have a carbon base and the carbon atom itself allows 4 opportunities to bond with other elements. The methane molecule described by a carbon atom at the center and 4 hydrogen atoms can take on the appearance of the tetrahedron with the carbon atom at the center. (see figure 11)

(figure available in print form)

methane molecule

> as a tetrahedron

Figure 11

Since all living things are carbon based, all other carbon compounds can be described with tetrahedrons.

The crystal structure of minerals found outdoors or on display in museums also exhibit the same kinds of patterns and systems. These crystals reflects the growth of the mineral based on the atomic structure of the mineral. We have reviewed the basic cubic, tetrahedral, and hexagonal patterns. The structure of crystal are described by these same terms. This is not to say that a perfect crystal structure can always be found, but traces of the systems can be identified if we have a good specimen. We can study a specimen of pyrite, or foolís gold, and see that the crystals are cubic. Halite, or rock salt, also exhibits cubic crystals.

Quartz, a mineral common to our area, can be seen in the form of hexagonal crystals while fluorite, a mineral that possesses the property of glowing in ultra-violet exhibits distinct tetrahedral crystals. These crystal formations are indications of the growth patterns of the minerals. In general tetragonal crystals are often long and slender or needle-like. They are characteristically four sided prisms or pyramids. Hexagonal crystals are generally column or prism-like with hexagonal cross sections, while the cubic system exhibits crystals that are very blocky or ball-like in appearance with many similar, symmetrical faces. Again, these same geometrical patterns appear in all of nature.

Students can walk about the city streets and find many examples of repeating patterns. Ornamentation on windows can take the form of square patterns. Tiling on floors generally take the form of hexagonal patterns or some other combination of squares and hexagons. The same patterns persist. The same patterns are dictated by space to appear in nature, in art, in mathematics and in architecture. There is a sense of cohesiveness, of some master plan, an order to our universe. It is evident in the world around us and is manifested in architecture. It will not take a great deal of effort to open the eyes of our students to the knowledge available to them. Their natural curiosity will take over once theyíve become aware of the 3-dimensional world around them.